barrels, max volume. lowest price

Ok, so this is a rather interesting problem. Question states:
An old method to decide the price of a wine barrel was decided like this. You stick a measuring stick from the tap hole S until it touches the the lid D. The lenght SD = d was used to decide the price of the barrel. The astronom kepler said that the best buy of a barrel is if h=2d/ (3)^1/2. without using derivatives.
And you have this picture of half av the barrel written as h/2 to point s. the hyptenus is the distance d. is kepler right?
So i want to solve the problem like this. I want to know the maximum volume and the minimum distance d (SD). and im thinking about using the derivative of a cylinder which is 2pi*r^2 * h = volume cylinder. and for swapping out h. So here is where i get stuck. i can use pythagoras theorem for half of the barrel. d^2 = h^2 + r^2. Now the problem is that i end up with two variables no matter what. And i feel that i dont want to use keplers formula since i want to find out the maximum value and compare it to his value.
All help is appreciated

Ok, so this is a rather interesting problem. Question states:
An old method to decide the price of a wine barrel was decided like this. You stick a measuring stick from the tap hole S until it touches the the lid D. The lenght SD = d was used to decide the price of the barrel. The astronom kepler said that the best buy of a barrel is if h=2d/ (3)^1/2. without using derivatives.
And you have this picture of half av the barrel written as h/2 to point s. the hyptenus is the distance d. is kepler right?
So i want to solve the problem like this. I want to know the maximum volume and the minimum distance d (SD). and im thinking about using the derivative of a cylinder which is 2pi*r^2 * h = volume cylinder. and for swapping out h. So here is where i get stuck. i can use pythagoras theorem for half of the barrel. d^2 = h^2 + r^2. Now the problem is that i end up with two variables no matter what. And i feel that i dont want to use keplers formula since i want to find out the maximum value and compare it to his value.
All help is appreciated

Please try to make this clearer, so spacing might help as would a diagram